Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

July 13, 2023 at 7:00am

#1052549

You Realize This Means War

How do you think the war's going? No, not that war. Or that one. This one:

5 Completely Embarrassing Moments in the War on Drugs

Link is to Cracked, so at least this should be entertaining.

For decades now, the American government has been fighting the “War on Drugs.”

I always figured it was the conservative response to the liberal "War on Poverty."

When a war has been going on for almost a half-century, it’s not usually because you’re winning.

But admitting that you lost is embarrassing.

Unless the goal was to horrendously overcrowd the nation’s prisons with nonviolent offenders. In which case, you showed them!

As incarcerated individuals can legally be slaves, perhaps that was a goal all along: cheap labor. You want murderers digging ditches? I didn't think so.

The term crashed onto the scene after it was proclaimed in a speech by actual criminal Richard Nixon.

An actual criminal who avoided prison.

This was in 1971, shortly before he would resign from office due to the Watergate scandal.

Um, no. Nixon was elected to a second term in 1972, and it wasn't until August of 74 that he resigned. Three years is a significant portion of a Presidential term.

Here are five of the most embarrassing products and occurrences since the War on Drugs started

Yes, it's a countdown.

5. DARE Doesn’t Work

Maybe one of the most famous anti-drug campaigns ever created was D.A.R.E., which is what I assume is a backronym of the unwieldy Drug Abuse Resistance Education.

Everyone I knew said it stood for Drugs Are Really Excellent.

D.A.R.E. probably had more of an effect on the graphic T-shirt business than the drug trade.

There was a period in there when dealers would wear those shirts so potential customers could identify them. Or so I've been told.

4. Legal Weed’s Success

We have been trying to get weed legalized for as long as I can remember. It's clearly not in the same category as other mind-altering substances, some of which are legal, in terms of potential negative effects, and can have actual benefits. The War on Drugs slammed the brakes on that effort... for a while.

The devil’s lettuce was, for ages, one of the chief bogeymen in the War on Drugs’ lore as a gateway drug and the leading cause of reefer madness.

The "gateway drug" argument is, and always was, absolute nonsense.

Not only that, but the legalization of weed has created a booming business and a positively shocking amount of tax revenue for the states in which it’s implemented.

"Wait, we can make money off this instead of spending money to try to stop it?"

3. Quadro Tracker

Okay, no idea what this is. Let's find out.

It was a small plastic device that claimed to be able to detect things like guns and drugs after you inserted the corresponding “frequency card” (which, of course, cost money). The location was then meant to be indicated by a metal antenna. Now, calling it a device at all might have been generous, given that it turned out to be just a hollow piece of plastic. Basically, police across the country dished out taxpayer dollars in a big way to buy drug-dowsing rods.

Normally, I think people who perpetrate scams should be locked up. In this case, however, they should build statues.

2. ”Fentanyl Overdose” Cop Footage

The stories continue to pile up of officers left lying panicked on the ground or looking like they can see directly into the face of death after contact such as brushing it off their uniform, or just being in a car where the drug was found.

This has always made me laugh. I don't remember when fentanyl seemed to suddenly burst onto the scene; it wasn't that long ago that I first heard of it. Sometime in the last ten years, maybe. This century, for sure. But it was only because I kept seeing stories like "Officer gets a nanogram of fentanyl on skin, drops dead."

Absorption through skin can be an actual thing for some drugs (it's how nicotine patches work, for example), but if it's that deadly, how can it be manufactured, delivered, and eventually used? Even assuming clean-room procedures. By the time the cops find it, it's already out in the open. Why doesn't it kill the dealers, too?

1. Admitting It Was B.S. in the First Place

This, really, is the most damning banana in the whole bunch:

Nixon’s domestic policy chief, John Ehrlichman, in an interview from Harper magazine in 1994, finally gave up the ghost on the true motives of declaring a War on Drugs. Even as admissions of guilt go, it was pretty stark.

He told the interviewer, “We knew we couldn’t make it illegal to be either against the war or Black (people), but by getting the public to associate the hippies with marijuana and Blacks with heroin, and then criminalizing both heavily, we could disrupt those communities. We could arrest their leaders, raid their homes, break up their meetings and vilify them night after night on the evening news. Did we know we were lying about the drugs? Of course, we did.”

You mean... the government can't be trusted? They'll just make shit up to discredit people they don't want having any power or influence? No way!

(Of course, Ehrlichman might have been lying, too...)

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